Deadline Miss Rates of Applications with
Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng
{sorma, petel, zebpe}@ida.liu.se
Department of Computer and Information Science
Linköping University, Sweden
Probability
Motivation
Expensive hardware
0% missed deadlines
Probability
Task execution time
Affordable hardware
<5% missed deadlines
Task execution time
Deadline Miss Rate Analysis of Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
2
Problem formulation, input
2s
graphs
 Task periods
 Task execution time probability
2s
density functions
4s
 Message transmission time
probability density functions
 Task and task graph deadlines
6s
miss 4%
 Mapping of tasks to processors and
messages to buses
10s
 Deadline miss ratio thresholds
Probability
 Task
10s
Task execution time
miss 2%
miss 10%
Deadline Miss Rate Analysis of Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
3
Problem formulation, output
miss 0%
miss 2%
 Deadline
miss ratios per
task and task graph
miss 0%
miss 2%
miss 5%
miss 0%
miss 7%
miss 3%
Deadline Miss Rate Analysis of Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
miss 3%
4
Outline
 For
monoprocessor systems, we found an exact solution based on
concurrent construction and analysis of the underlying generalized
semi-Markov process
[Manolache et al. “Memory and Time-Efficient Schedulability Analysis
of Task Graphs with Stochastic Execution Time”, ECRTS-01]
 The solution is theoretically applicable to multiprocessor systems, but
practically to only very small ones, because of complexity
Solutions based on approximation
1. Execution time PDF approximation
2. Independence assumption among various
random variables
Deadline Miss Rate Analysis of Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
5
Execution Time PDF Approximation
Coxian approximation-based
Task graphs
Modelling
Approximation
GSPN
Coxian distribs
CTMC constr.
CTMC
Results
Analysis
Deadline Miss Rate Analysis of Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
7
Application modelling (1)
A
E
B
C
F
D
Deadline Miss Rate Analysis of Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
8
Application modelling (2)
A
B
C
D
F
C
F
probab
A
E
D
B
E
Firing delay equals
execution time
firing delay
Deadline Miss Rate Analysis of Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
9
Coxian approximation-based
Task graphs
Modelling
Approximation
GSPN
Coxian distribs
CTMC constr.
CTMC
Results
Analysis
Deadline Miss Rate Analysis of Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
10
CTMC construction (1)
X, Y
X, Y
X
GSMP
Approximation of the GSMP
X
Approximation of X
Deadline Miss Rate Analysis of Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
11
CTMC construction (2)
 The global generator of the Markov chain becomes
then
M  (  Aj )  I 
jen
  (  I j )  Bi  (  I j )  D j 
ien
j i ,
jen
j i ,
jen
 M is expressed in terms of small matrices and can be
generated on the fly – memory savings
Deadline Miss Rate Analysis of Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
12
Analysis time vs. number of tasks
Deadline Miss Rate Analysis of Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
13
Analysis time vs. number of procs
Deadline Miss Rate Analysis of Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
14
Growth with number of stages
Deadline Miss Rate Analysis of Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
15
Accuracy
Accuracy vs analysis complexity compared to the
exact approach
Stages
2
3
4
5
Relative error 8.7% 4.1% 1.04% 0.4%
Deadline Miss Rate Analysis of Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
16
Independence Assumption-Based
Approximation
Independence assumption-based
 Faster
and approximate analysis for multiprocessor systems
[ICCAD 2002]
 However it is still too slow to be plugged into an optimization loop

 Analysis
complexity is reduced by two means:
 Task start and finish times are approximated with discrete values
 Two types of dependencies between some random variables are
neglected
Deadline Miss Rate Analysis of Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
18
Independence of predecessors
A
Z
Y
X
Z
Y
X
Z
Y
X
P(X>max(Y, Z)) = P(X>Y)  P(X>Z)
Deadline Miss Rate Analysis of Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
19
Load-arrival time independence
A
C
B
A
B
C
Time
P(LC(t)) = P(LC(t)|AC<t)
Deadline Miss Rate Analysis of Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
20
Approximation effects
Deadline Miss Rate Analysis of Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
21
Experimental results
Deadline Miss Rate Analysis of Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
22
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Conclusions
 Two
approaches for obtaining approximations of deadline miss
ratios
 Based on the approximation of the ETPDF by Coxian
distributions
 Efficient scheme to store the underlying stochastic
process and to construct it on the fly
 Based on independence assumptions among random variables
 Both
approaches provide the possibility to trade analysis speed and
memory demand for analysis accuracy
Deadline Miss Rate Analysis of Applications with Stochastic Task Execution Times
Sorin Manolache, Petru Eles, Zebo Peng – Linkoping University, Sweden
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